Probability Concepts and Rules: Chapters 11 & 12 Study Notes

Probability Concepts

Random Experiments and Sample Space

Probability theory begins with the study of random experiments, where the outcome cannot be predicted with certainty. The sample space (S) is the set of all possible outcomes of a random experiment.

• Sample Space (S): The complete set of possible outcomes.

  • Example: Tossing a coin, S = {Head, Tail}

  • Example: Rolling a die, S = {1, 2, 3, 4, 5, 6}

• Event: An event is a subset of the sample space, representing one or more outcomes.

  • Example: Tossing a head is an event. Tossing two heads in two tosses is also an event.

Probability of an Event

The probability of an event A, denoted P(A), quantifies the likelihood that event A will occur.

• Properties of Probability:

  • $0 \leq P(A) \leq 1$

  • $P(A) = 0$ means event A is impossible.

  • $P(A) = 1$ means event A is certain.

  • The larger $P(A)$, the more likely event A will occur.

• Sum of Probabilities: The sum of the probabilities of all non-overlapping (mutually exclusive) events in the sample space equals 1.

  • Example: Rolling a fair die, each outcome has probability $\frac{1}{6}$, and $\frac{1}{6} + \frac{1}{6} + \frac{1}{6} + \frac{1}{6} + \frac{1}{6} + \frac{1}{6} = 1$.

Terminology and Notation

Complement, Union, and Intersection of Events

Understanding how events relate to each other is essential in probability.

• Complement of an Event: The event that A does not occur, denoted $A^c$.

• Union of Events (A or B): The event that either A or B or both occur. Denoted $A \cup B$.

• Intersection of Events (A and B): The event that both A and B occur together. Denoted $A \cap B$.

Disjointness and Independence

• Disjoint (Mutually Exclusive) Events: Two events are disjoint if they cannot occur together. $A \cap B = \emptyset$.

• Independence of Events: Two events are independent if the occurrence of one does not affect the probability of the other.

• Independence of Trials: In repeated experiments (e.g., coin tosses), trials are independent if the outcome of one does not influence the others.

Probability Rules

Basic Probability Rules

• Probability of the Sample Space: $P(S) = 1$

• Complement Rule: $P(A^c) = 1 - P(A)$

• Addition Rule for Disjoint Events: If A and B are disjoint, $P(A \text{ or } B) = P(A) + P(B)$

• General Addition Rule: If A and B are not disjoint, $P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$

• Equally Likely Outcomes: If all outcomes are equally likely, $P(A) = \frac{\text{number of outcomes in event A}}{\text{total number of outcomes in the sample space}}$

Conditional Probability

Conditional probability measures the probability of event B occurring given that event A has occurred.

• Conditional Probability Formula:

  • $P(B|A) = \frac{P(A \text{ and } B)}{P(A)}$

Multiplication Rule

• Independent Events: If A and B are independent, $P(A \text{ and } B) = P(A) \times P(B)$

• Dependent Events: If A and B are dependent, $P(A \text{ and } B) = P(A) \times P(B|A)$ or $P(B) \times P(A|B)$

Testing Independence

To determine if two events A and B are independent, check if any of the following holds:

• $P(A) = P(A|B)$

• $P(B) = P(B|A)$

• $P(A \text{ and } B) = P(A) \times P(B)$

Exercises and Applications

Disjoint and Independent Events

• Disjoint Events Example:

  • Rolling a fair die: A = rolling an even number, B = rolling an odd number (disjoint).

  • Randomly selecting a family with two children: A = first kid is a boy, B = second kid is a boy (not disjoint).

• Independent Events Example:

  • Tossing a coin twice: A = head in first toss, B = tail in second toss (independent).

  • Drawing two cards from a deck with replacement: A = first card is a spade, B = second card is a five (independent).

  • Drawing two cards from a deck without replacement: A = first card is a spade, B = second card is a spade (not independent).

Probability in Real-World Contexts

• Ownership Example: In a college, 65% of students own a car, 82% own a computer, and 55% own both. Find the probability that a randomly chosen student owns neither a car nor a computer.

  • Let A = owns a car, B = owns a computer.

  • Use the complement and addition rules to solve.

• Student Location Table: The following table summarizes the distribution of students by location and type:

	Vancouver (V)	Richmond (R)	Burnaby or Coquitlam (B)	Total
Domestic (D)	18	10	7	35
International (I)	24	15	6	45
Total	42	25	13	80

• Example Applications:

  • Given a student lives in Richmond, find the probability they are domestic.

  • Given a student is international, find the probability they do not live in Vancouver.

  • Find the probability a student is international or lives in Vancouver.

  • Define events and test independence (e.g., international student vs. living in Burnaby/Coquitlam).

  • Find probabilities for selections with and without replacement.

Summary Table: Probability Rules

Rule	Formula (LaTeX)	When to Use
Complement Rule	$P(A^c) = 1 - P(A)$	Finding probability that event A does not occur
Addition Rule (Disjoint)	$P(A \text{ or } B) = P(A) + P(B)$	Events A and B are disjoint
Addition Rule (General)	$P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$	Events A and B may overlap
Multiplication Rule (Independent)	$P(A \text{ and } B) = P(A) \times P(B)$	Events A and B are independent
Multiplication Rule (General)	$P(A \text{ and } B) = P(A) \times P(B|A)$	Events A and B may be dependent
Conditional Probability	$P(B|A) = \frac{P(A \text{ and } B)}{P(A)}$	Probability of B given A has occurred

Additional info:

• These notes cover foundational probability concepts, including sample spaces, events, probability rules, conditional probability, independence, and applications to real-world scenarios and exercises.